### Reflection in the x-Axis

We say that a line or curve has been *reflected in the x-axis* when each point on it has been effectively “flipped” across the x -axis. If we imagine that the x -axis is a mirror, the reflection is what would appear in the mirror. The reflection in the x -axis of a graph thus looks like an upside down version of the original graph. Each point on the new graph is the same distance from the x -axis as the corresponding point on the original graph was, except on the other side of the x -axis.

If a point on the original graph is (a,b) , then upon reflection in the x -axis, this point becomes (a,-b). That means that if we have a function f(x) , then the reflection of its graph in the x -axis is -f(x) . If we start out with the function y=x^2+3 , for example, then reflecting this in the x -axis gives us y=-x^2-x , as the following graph shows:

### Reflection in the y-Axis

We say that a line or curve has been *reflected in the y-axis* when each point on it has been effectively “flipped” across the y -axis. If we imagine that the y -axis is a mirror, the reflection is what would appear in the mirror. The reflection in the y -axis of a graph thus looks like an upside down version of the original graph. Each point on the new graph is the same distance from the y -axis as the corresponding point on the original graph was, except on the other side of the y -axis.

If a point on the original graph is (a,b) , then upon reflection in the y -axis, this point becomes (-a,b). That means that if we have a function f(x) , then the reflection of its graph in the y -axis is f(-x) . If we start out with the function y=(x-4)^3 , for example, then reflecting this in the x -axis gives us y=(-x-4)^3 , which we could rewrite as y=-(x+4)^3 . To help visualize this, check out the graph below.

A *rigid transformation* is a modification to a graph that does not distort the shape of the graph but simply moves it. Translations (shifts) and reflections are rigid transformations.

A *nonrigid transformation* is a modification to a graph that distorts the shape of the graph. This can happen through stretching or shrinking the graph either vertically or horizontally, although we will only deal with vertical shrinks and stretches in this course.